🎓 Lesson 10
D5
Swing Equation & Equal Area Criterion
The Swing Equation describes how a generator’s rotor speed changes when power imbalances occur, and the Equal Area Criterion is a simple graphical method to check if the generator will regain stable operation after a disturbance.
🎯 Learning Objectives
- ✓ Derive and linearize the Swing Equation for small-signal stability analysis
- ✓ Apply the Equal Area Criterion to assess transient stability of a single-machine infinite-bus system
- ✓ Calculate critical clearing angle and time for a three-phase fault using EAC
- ✓ Explain the physical meaning of accelerating and decelerating areas in terms of kinetic energy exchange
- ✓ Analyze how system parameters (inertia, fault location, pre-fault loading) affect stability margins
📖 Why This Matters
In mining operations, large electric drives (e.g., rope shovels, conveyor systems, SAG mills) rely on stable grid-connected synchronous motors and generators. A sudden fault—like a short circuit near a substation feeding a mine’s processing plant—can cause rotor angle swings that, if unchecked, lead to loss of synchronism, equipment tripping, and costly production stoppages. Understanding the Swing Equation and Equal Area Criterion allows blasting and mining engineers to collaborate with power system specialists in specifying protection settings, validating grid interface studies, and designing resilient electrified mine infrastructure.
📘 Core Principles
The Swing Equation originates from Newton’s second law applied to rotating mass: J·d²δ/dt² = Pₘ − Pₑ, where δ is rotor angle, Pₘ is mechanical power input, and Pₑ is electrical power output. Under per-unit normalization, it simplifies to M·d²δ/dt² = Pₘ − Pₑ, where M = 2H/ω₀ (H = inertia constant in MJ/MVA, ω₀ = synchronous speed in rad/s). The Equal Area Criterion emerges by integrating the Swing Equation with respect to δ, yielding ∫(Pₘ − Pₑ)dδ = ½M(dδ/dt)² + C — i.e., net area under the P–δ curve equals change in kinetic energy. Stability requires the system to decelerate enough after fault clearance to halt angular separation before δ exceeds 180°. Key assumptions include constant Pₘ, lossless transmission, and rigid machine model — all reasonable for first-pass mining grid stability screening.
📐 Swing Equation & Equal Area Criterion
The normalized Swing Equation governs rotor dynamics; the Equal Area Criterion provides a direct stability test without solving the differential equation. For a single-machine infinite-bus system, stability holds if A₁ = A₂, where A₁ is the accelerating area (pre-clearing) and A₂ is the decelerating area (post-clearing) on the P–δ diagram.
💡 Worked Example
Problem: A 500 MVA synchronous generator (H = 4.0 s) operates at δ₀ = 30° (π/6 rad) delivering 0.8 p.u. power to an infinite bus. A three-phase fault occurs at the generator terminals, reducing Pₑ to 0. Fault is cleared at δ_c = 60° (π/3 rad), after which Pₑ restores to Pₑ = 1.5 sin δ. Find critical clearing angle δ_cr.
1.
Step 1: Pre-fault power: Pₘ = Pₑ₀ = 0.8 = P_max₀ sin δ₀ → P_max₀ = 0.8 / sin(30°) = 1.6 p.u.
2.
Step 2: During fault: Pₑ = 0 ⇒ accelerating area A₁ = ∫_{δ₀}^{δ_c} (Pₘ − 0) dδ = Pₘ(δ_c − δ₀) = 0.8 × (π/3 − π/6) = 0.8 × π/6 ≈ 0.4189 rad·p.u.
3.
Step 3: Post-fault Pₑ = 1.5 sin δ; equilibrium at Pₘ = Pₑ ⇒ sin δ_∞ = 0.8 / 1.5 ≈ 0.5333 ⇒ δ_∞ ≈ 32.2°, but stability limit occurs when A₁ = A₂ = ∫_{δ_c}^{δ_cr} (0 − 1.5 sin δ) dδ = 1.5(cos δ_cr − cos δ_c). Set equal: 0.8(δ_cr − δ₀) = 1.5(cos δ₀ − cos δ_cr). Solve numerically → δ_cr ≈ 105.2° (1.836 rad).
4.
Step 4: Critical clearing time t_cr ≈ √(2M·δ_cr / Pₘ) for small angles (approximate) or integrate d²δ/dt² = (Pₘ − 0)/M → t_cr = √(2M(δ_cr − δ₀)/Pₘ) = √(2×(2×4.0/2π)×(1.836−0.524)/0.8) ≈ 0.27 s.
Answer:
Critical clearing angle δ_cr ≈ 105°, corresponding to ~0.27 s — well within typical relay+breaker clearing time of 0.15–0.30 s. Thus, the system is transiently stable for this fault scenario.
🏗️ Real-World Application
At Newmont’s Boddington Gold Mine (Western Australia), a 2021 grid interaction study assessed transient stability following a 132 kV feeder fault near the primary substation supplying the crushing and grinding plant. Using the classical model and EAC, engineers determined that existing overcurrent relays (set for 0.22 s clearing) were adequate for the worst-case bolted fault, but required coordination review when a new 40 MW battery energy storage system (BESS) was added — as BESS injection altered post-fault P–δ characteristics and reduced decelerating area. The EAC provided rapid go/no-go screening before committing to full EMTP simulations.
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